Analysis on radial complex-valued functions of
p-adic arguments
Anatoly N. KochubeiInstitute of Mathematics,
National Academy of Sciences of Ukraine,Tereshchenkivska 3, Kiev, 01024 Ukraine,
E-mail: [email protected]
Preliminaries
1. p-Adic numbers
Let p be a prime number. The �eld of p-adic numbers is the
completion Qp of the �eld Q of rational numbers, with respect to
the absolute value |x |p de�ned by setting |0|p = 0,
|x |p = p−ν if x = pνm
n,
where ν,m, n ∈ Z, and m, n are prime to p. Example: |p|p = p−1.
Qp is a locally compact topological �eld.
Note that by Ostrowski's theorem there are no absolute values on
Q, which are not equivalent to the �Euclidean� one, or one of | · |p.We denote Zp = {x ∈ Qp : |x |p ≤ 1}. Zp, as well as all balls in
Qp, is simultaneously open and closed.
The absolute value |x |p, x ∈ Qp, has the following properties:
|x |p = 0 if and only if x = 0;
|xy |p = |x |p · |y |p;
|x + y |p ≤ max(|x |p, |y |p).
The latter property called the ultra-metric inequality (or the
non-Archimedean property) implies the total disconnectedness of
Qp in the topology determined by the metric |x − y |p, as well asmany unusual geometric properties (Example: two balls either do
not intersect, or one of them is contained in another). Note also
the following consequence of the ultra-metric inequality:
|x + y |p = max(|x |p, |y |p) if |x |p 6= |y |p.
The absolute value |x |p takes the discrete set of non-zero values
pN , N ∈ Z. If |x |p = pN , then x admits a (unique) canonical
representation
x = p−N(x0 + x1p + x2p
2 + · · ·),
where x0, x1, x2, . . . ∈ {0, 1, . . . , p − 1}, x0 6= 0. The series
converges in the topology of Qp. For example,
−1 = (p − 1) + (p − 1)p + (p − 1)p2 + · · · , | − 1|p = 1.
The canonical representation shows the hierarchical structure of Qp.
Figure: Structure of the p-adic tree.
Khrennikov et al (2016) - p-adic model of a porous medium.
Applications:
- leakage of a liquid through a porous medium;
- physics of hierarchical structures (for example, dynamics of
protein molecules).
Mathematical developments:
Since rational numbers are used in almost all branches of
mathematics, Ostrowski's theorem (1916) has divided mathematics
into two parts, the classical one and the non-Archimedean one,
based on p-adic numbers.
By this time, there exist non-Archimedean counterparts of almost
all branches of mathematics.
2. Local �elds. Let K be a non-Archimedean local �eld, that is a
non-discrete totally disconnected locally compact topological �eld.
It is well known that K is isomorphic either to a �nite extension of
the �eld Qp of p-adic numbers (if K has characteristic 0), or to the
�eld of formal Laurent series with coe�cients from a �nite �eld, if
K has a positive characteristic.
Any local �eld K is endowed with an absolute value | · |K , such that
|x |K = 0 if and only if x = 0, |xy |K = |x |K · |y |K ,|x + y |K ≤ max(|x |K , |y |K ). Denote O = {x ∈ K : |x |K ≤ 1},P = {x ∈ K : |x |K < 1}. O is a subring of K , and P is an ideal in
O containing such an element β that P = βO. The quotient ring
O/P is actually a �nite �eld; denote by q its cardinality. We will
always assume that the absolute value is normalized, that is
|β|K = q−1. The normalized absolute value takes the values qN ,N ∈ Z. Note that for K = Qp we have β = p and q = p; thep-adic absolute value is normalized.
Denote by S ⊂ O a complete system of representatives of the
residue classes from O/P . Any nonzero element x ∈ K admits the
canonical representation in the form of the convergent series
x = β−n(x0 + x1β + x2β
2 + · · ·)
where n ∈ Z, |x |K = qn, xj ∈ S , x0 /∈ P . For K = Qp, one may
take S = {0, 1, . . . , p − 1}.The additive group of any local �eld is self-dual, that is if χ is a
�xed non-constant complex-valued additive character of K , then
any other additive character can be written as χa(x) = χ(ax),x ∈ K , for some a ∈ K . Below we assume that χ is a rank zero
character, that is χ(x) ≡ 1 for x ∈ O, while there exists such an
element x0 ∈ K that |x0|K = q and χ(x0) 6= 1.
Fourier transform:
f̃ (ξ) =
∫K
χ(xξ)f (x) dx , ξ ∈ K ,
Inverse transform:
f (x) =
∫K
χ(−xξ)f̃ (ξ) dξ.
Test functions from D(K ): locally constant functions with compact
supports.
D′(K ) - Bruhat-Schwartz distributions.
Vladimirov operator:
(Dαϕ) (x) = F−1 [|ξ|αK (F(ϕ))(ξ)] (x), α > 0,
ϕ ∈ D(K ).The operator Dα can also be represented as a hypersingular
integral operator:
(Dαϕ) (x) =1− qα
1− q−α−1
∫K
|y |−α−1K [ϕ(x − y)− ϕ(x)] dy .
This expression makes sense for wider classes of functions.
As an operator in L2, Dα is selfadjoint, nonnegative with pure point
spectrum {qαN , N ∈ Z} of in�nite multiplicity.
Right inverse (α > 0):
(D−αϕ
)(x) = (fα∗ϕ)(x) =
1− q−α
1− qα−1
∫K
|x−y |α−1K ϕ(y) dy , ϕ ∈ D(K ), α 6= 1,
and (D−1ϕ
)(x) =
1− q
q log q
∫K
log |x − y |Kϕ(y) dy .
Then DαD−α = I on D(K ), if α 6= 1. This property remains valid
on Φ(K ) also for α = 1. Here
Φ(K ) =
ϕ ∈ D(K ) :
∫K
ϕ(x) dx = 0
.
is the so-called Lizorkin space.
Vladimirov Operator on Radial Functions
a) Radial eigenfunctions
Let u(x) = ψ(|x |) ∈ L2(K ),
Dαu = λu, λ = qαN , N ∈ Z,
and u is not identically zero. Then
u(x) =
cqN(1− q−1), if |x | ≤ q−N ;
−cqN−1, if |x | = q−N+1;
0, if |x | > q−N+1.
It is shown that u ∈ Φ(K ).The only radial eigenfunction u with u(0) = 1 (an analog of the
function t → e−λt , t ∈ R) corresponds to c = q−N(1− q−1)−1.On the other hand, if u(0) = 0, then c = 0.
This is the cornerstone of the theory of non-Archimedean wave
equation.
b) Explicit formula
LemmaIf a function u = u(|x |K ) is such that
m∑k=−∞
qk∣∣∣u(qk)
∣∣∣ <∞, ∞∑l=m
q−αl∣∣∣u(ql)
∣∣∣ <∞,for some m ∈ Z, then for each n ∈ Z the hypersingular integral for
Dαϕ with ϕ(x) = u(|x |K ) exists for |x |K = qn, depends only on
|x |K , and
(Dαu)(qn) = dα
(1− 1
q
)q−(α+1)n
n−1∑k=−∞
qku(qk)
+ q−αn−1qα + q − 2
1− q−α−1u(qn) + dα
(1− 1
q
) ∞∑l=n+1
q−αlu(ql)
where dα =1− qα
1− q−α−1.
The regularized integral
(Iαϕ)(x) = (D−αϕ)(x)− (D−αϕ)(0).
This is de�ned initially for ϕ ∈ D(K ). In fact,
(Iαϕ)(x) =1− q−α
1− qα−1
∫|y |K≤|x |K
(|x − y |α−1K − |y |α−1K
)ϕ(y) dy , α 6= 1,
and
(I 1ϕ)(x) =1− q
q log q
∫|y |K≤|x |K
(log |x − y |K − log |y |K )ϕ(y) dy .
In contrast to D−α, the integrals are taken, for each �xed x ∈ K ,
over bounded sets.
MAIN LEMMA
Suppose that
m∑k=−∞
max(qk , qαk
) ∣∣∣u(qk)∣∣∣ <∞, if α 6= 1,
m∑k=−∞
|k |qk∣∣∣u(qk)
∣∣∣ <∞, if α = 1,
for some m ∈ Z. Then Iαu exists, it is a radial function, and for
any x 6= 0,
(Iαu)(|x |K ) = q−α|x |αKu(|x |K )
+1− q−α
1− qα−1
∫|y |K<|x |K
(|x |α−1K − |y |α−1K
)u(|y |K ) dy , α 6= 1,
and
(I 1u)(|x |K ) = q−1|x |Ku(|x |K )
+1− q
q log q
∫|y |K<|x |K
(log |x |K − log |y |K ) u(|y |K ) dy .
Proposition (�right inverse�)
Suppose that for some m ∈ Z,
m∑k=−∞
max(qk , qαk
) ∣∣∣v(qk)∣∣∣ <∞, ∞∑
l=m
∣∣∣v(ql)∣∣∣ <∞,
if α 6= 1, and
m∑k=−∞
|k |qk∣∣∣v(qk)
∣∣∣ <∞, ∞∑l=m
l∣∣∣v(ql)
∣∣∣ <∞,if α = 1. Then there exists (DαIαv) (|x |K ) = v(|x |K ) for any
x 6= 0.
Proposition (�left inverse�)
Suppose that u(0) = 0,
|u(qn)| ≤ Cqdn, n ≤ 0;
|u(qn)| ≤ Cqhn, n ≥ 0,
where d > max(0, α− 1), 0 ≤ h < α, and h < α− 1, if α > 1.Then the function w = Dαu satis�es, for any m ∈ Z, theinequalities
m∑k=−∞
max(qk , qαk)∣∣∣w(qk)
∣∣∣ <∞, ∞∑l=m
∣∣∣w(ql)∣∣∣ <∞, α 6= 1;
m∑k=−∞
|k | · qk∣∣∣w(qk)
∣∣∣ <∞, ∞∑l=m
|l | ·∣∣∣w(ql)
∣∣∣ <∞, α = 1.
Moreover, IαDαu = u.
Corollary
Let v = v0 + u where v0 is a constant, u satis�es the conditions of
the above Proposition. Then IαDαv = v − v0.
EXAMPLE: the simplest Cauchy problem
Dαu(|x |K ) = f (|x |K ), u(0) = 0,
where f is a continuous function, such that
∞∑l=m
∣∣∣f (ql)∣∣∣ <∞, if α 6= 1, or
∞∑l=m
l∣∣∣f (ql)
∣∣∣ <∞, if α = 1.
The unique strong solution is u = Iαf . Therefore on radial functions,
the operators Dα and Iα behave like the Caputo-Dzhrbashyan fractional
derivative and the Riemann-Liouville fractional integral of real analysis.
(COUNTER)-EXAMPLE: Let f (|x |K ) ≡ 1, x ∈ K . Then
(Iαf ) (|x |K ) ≡ 0.
THE CAUCHY PROBLEM
a) Local solvability. Consider the equation
(Dαu) (|t|K ) = f (|t|K , u(|t|K )), 0 6= t ∈ K , (1)
with the initial condition
u(0) = u0 (2)
where the function f : qZ × R→ R satis�es the conditions
|f (|t|K , x)| ≤ M; (3)
|f (|t|K , x)− f (|t|K , y)| ≤ F |x − y |, (4)
for all t ∈ K , x , y ∈ R.
With the problem (1)-(2) we associate the integral equation
u(|t|K ) = u0 + Iαf (| · |K , u(| · |K ))(|t|K ). (5)
Note that, by the de�nition of Iα, in order to compute (Iαϕ)(|t|K )for |t|K ≤ qm (m ∈ Z), one needs to know the function ϕ in the
same ball |t|K ≤ qm. Therefore local solutions of the equation (5)
make sense, in contrast to solutions of (1).
We call a solution u of (5), if it exists, a mild solution of the
Cauchy problem (1)-(2). By the above Corollary, a solution u of
(1)-(2), such that u− u0 satis�es the conditions of Proposition, is a
mild solution.
TheoremUnder the assumptions (3),(4), the problem (1)-(2) has a unique
local mild solution, that is the integral equation (5) has a solution
u(|t|K ) de�ned for |t|K ≤ qN where N ∈ Z is su�ciently small, and
another solution u(|t|K ), if it exists, coincides with u for |t|K ≤ qN
where N ≤ N.
b) Extension of solutions. Let us study the possibility to continue
the local solution constructed in Theorem 1 to a solution of the
integral equation (5) de�ned for all t ∈ K .
Suppose that the conditions of Theorem 1 are satis�ed, and we
obtained a local solution u(|t|K ), |t|K ≤ qN , N ∈ Z. Let α 6= 1. Inorder to �nd a solution for |t|K = qN+1, we have to solve the
equation
u(qN+1) = u0 + v(N)0 + qαN f (qN+1, u(qN+1)) (6)
where
v(N)0 =
1− q−α
1− qα−1
∫|y |K≤qN
(q(N+1)(α−1) − |y |α−1K
)f (|y |K , u(|y |K )) dy .
(7)
is a known constant. A similar equation can be written for α = 1.Then the above procedure, if it is successful, is repeated for all
l > N.
TheoremSuppose that the conditions of local existence theorem are satis�ed,
as well as the following Lipschitz condition:
|f (ql , x)− f (ql , y)| ≤ Fl |x − y |, x , y ∈ R, l ∈ Z, (8)
where 0 < Fl < q−αl for each l ∈ Z. Then a local solution of the
equation (5) admits a continuation to a global solution de�ned for
all t ∈ K .
c) From an integral equation to a di�erential one. Let usstudy conditions, under which the above continuation procedure
leads to a solution of the problem (1)-(2). As before, we assume
the conditions (3),(4) and (8). In addition, we will assume that
|f (ql , x)| ≤ Cq−βl , l ≥ 1, for all x ∈ R, (9)
where β > α.
TheoremUnder the assumptions (3),(4), (8) and (9), the mild solution
obtained by the iteration process with subsequent continuation,
satis�es the equation (1).
Some Publications
1. A. N. Kochubei, A non-Archimedean wave equation, Pacif. J. Math. 235
(2008), 245�261.
2. A. N. Kochubei, Radial solutions of non-Archimedean pseudodi�erentialequations, Pacif. J. Math. 269 (2014), 355�369.
3. A. N. Kochubei, Nonlinear pseudo-di�erential equations for radial realfunctions on a non-Archimedean �eld. J. Math. Anal. Appl. 483 (2020),no. 1, Article 123609.
4. A. N. Kochubei, Non-Archimedean Radial Calculus: Volterra Operatorand Laplace Transform, arXiv 2005.11166.
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